Inflation and dark energy from f(r) gravity

Transcription

1 Prepared for submissio to JCAP Iflatio ad dark eergy from f(r) gravity arxiv: v2 [hep-th] 22 Dec 2014 Micha l Artymowski a Zygmut Lalak b a Departmet of Physics, Beijig Normal Uiversity Beijig , Chia b Istitute of Theoretical Physics, Faculty of Physics, Uiversity of Warsaw ul. Hoża 69, Warszawa, Polad artymowski@bu.edu.c, Zygmut.Lalak@fuw.edu.pl Abstract. The stadard Starobisky iflatio has bee exteded to the R + αr βr 2 model to obtai a stable miimum of the Eistei frame scalar potetial of the auxiliary field. As a result we have obtaied obtai a scalar potetial with o-zero value of residual vacuum eergy, which may be a source of Dark Eergy. Our results ca be easily cosistet with PLANCK or BICEP2 data for appropriate choices of the value of. Keywords: Iflatio, f(r) theory, cosmic acceleratio, BICEP2 results ArXiv eprit:

2 Cotets 1 Itroductio 1 2 The R + αr model The geeratio of primordial ihomogeeities 6 3 The R + αr βr 2 model Coditios for the f(r) dark eergy 9 4 Numerical aalysis of the dark eergy model 11 5 Coclusios 11 6 Ackowledgemets 13 A Oscillatios aroud the miimum 13 1 Itroductio Iflatioary sceario for the Early Uiverse has become oe of the stadard assumptios i theoretical model buildig. It ca accout aturally for the wealth of preset observatioal data. The existece of rapid expasio stage i the history of the Uiverse ca give us a atural explaatio of the homogeeity, flatess ad horizo problems, ad provides a explaatio of the process of seedig large-scale structure ad temperature aisotropies of the cosmic microwave backgroud radiatio (see refs. [1 3] for reviews). Oe of the first iflatioary models was the Starobisky iflatio [4], based o the f(r) = R + R 2 /6M 2 Lagragia desity. I this theory durig iflatio the R 2 term domiates, which provides a stage of the de-sitter-like evolutio of space-time. The iflatioary potetial has a stable miimum, which allows for the graceful exit, reheatig ad good low-eergy limit of the theory. However, the recet observatioal data from the BICEP2 experimet [5, 6] suggest sigificat amout of primordial gravitatioal waves, which disfavours the Starobisky iflatio. As show i [7], a small modificatio of the f(r) fuctio, amely f(r) = R +αr, ca geerate iflatio which fits the BICEP2 data. This model has already bee partially discussed i [8]. Differet modificatios of the Starobisky model were also discussed i Ref. [9, 10]. O the other had the series of experimets [11 13] covicigly suggests the existece of the so-called Dark Eergy (DE) with barotropic parameter close to 1. Oe of the possible sources of DE may be a o-zero vacuum eergy of a scalar field, which i priciple ca be the f(r) theory [8, 14, 15] or the Bras-Dicke field. I this paper we demostrate how to exted the f(r) = R + αr model to combie successfully both features of the observable Uiverse i a sigle framework: we show how to obtai successful iflatio ad the o-zero residual value of the Ricci scalar i a extesio of the Starobisky model. Namely, we use the f(r) = αr βr 2 Lagragia desity, 1

3 which provides iflatio from the αr ad a stable miimum of the scalar potetial of a auxiliary field. The potetial has a o-zero vacuum eergy desity, which is the source of the DE i the late-times evolutio. Let us ote that some iflatioary potetials, which geerate DE (motivated by modified theories of gravity) were already itroduced i the Ref. [16] I the followig draft we use the covetio 8πG = M 2 pl = 1, where M pl GeV is the reduced Plack mass. The structure of this paper is as follows. I the Sec. 2 we aalyse the f(r) = R + αr model: we discuss aalytic solutios, parameters of iflatio ad primordial ihomogeeities. I the Sec. 3 we geeralise this model to f(r) = R + αr βr 2, which leads to the ozero vacuum eergy of the Eistei frame potetial. I the Sec. 4 we discuss umerical study of the evolutio of the model with dust modellig the cotributio of matter fields to eergy desity. Fially, we coclude i the Sec The R + αr model Let us cosider a f(r) theory i the flat FRW space-time with the metric tesor of the form of ds 2 = dt 2 + a(t) 2 (d x) 2. The, the Friedma equatios become 3F H 2 = (F R f)/2 3HF + ρ M, (2.1) 2F Ḣ = F HF + (ρ M + P M ), (2.2) where F (R) = df dr ad ρ M ad P M are eergy desity ad pressure of matter fields respectively. The actio of f(r) model ca be rewritte as S = d 4 x [ ] 1 g 2 ϕr U(ϕ) + L m, (2.3) where ϕ = F (R), U(ϕ) = 1 (RF f), (2.4) 2 which meas that the f(r) gravity ca be expressed as a Bras-Dicke theory i Jorda frame with ω BD = 0. I the Jorda frame the auxiliary field ϕ is o-miimally coupled to gravity, which creates a deviatio from the Geeral Relativity (GR) frame. The first Friedma equatio ad cotiuity equatios are the followig oes ϕ + 3H ϕ (ϕu ϕ 2U) = 1 3 (ρ M 3p M ), (2.5) ( 3 H + ϕ ) 2 = 3 ϕ 2 2ϕ 4 ϕ 2 + U ϕ + ρ M ϕ, (2.6) ρ M + 3H(ρ M + P M ) = 0. (2.7) where U ϕ = du dϕ. Let us ote that U may be iterpreted as a eergy desity, but U ϕ is ot a effective force i the Eq. (2.5). Oe ca defie the effective potetial ad its derivative - 2

4 the effective force, by 2 U eff = 3 (ϕu ϕ 2U)dϕ ϕ 1 3 (ρ M 3p M ) + C, F eff := du eff dϕ, (2.8) where C is ukow costat of itegratio. The effective potetial shall be iterpreted as a source of a effective force, but ot as a eergy desity. The gravitatioal part of the actio may obtai its caoical (miimally coupled to ϕ) form after trasformatio to Eistei frame which gives the actio of the form of g µν = ϕg µν, d t = ϕdt, ã = ϕa (2.9) S = d 4 x g 1 2 R 3 4 ( ) 2 ϕ U(ϕ) ϕ ϕ 2, (2.10) where is the derivative with respect to x µ. I order to obtai the caoical kietic term for ϕ let us use the Eistei frame scalar field φ 3 φ = log ϕ. (2.11) 2 The actio i terms of g µν ad φ looks as follows S = d 4 x [ 1 g 2 R 1 ( ) ] 2 φ V (φ), (2.12) 2 where V = (F R f) 2F 2 = U(ϕ) ϕ 2. (2.13) ϕ=ϕ(φ) Let us ote that (ϕu ϕ 2U) from the Eq. (2.5) ca be expressed as ϕ 3 V ϕ, so the miimum of V shall also be the miimum of the effective potetial i the Jorda frame. I fact, all importat features of the potetial, like existece of miima ad barriers betwee them, which determie the evolutio of the field i the Eistei frame are reflected i the evolutio of the field i the Jorda frame. I further parts of this draft we will refer to the Eistei frame potetial, eve though we cosider the Jorda frame as the primordial (or defiig) oe. Sice all of the aalysis performed i this draft is classical, descriptios i both frames give the same physical results. However, the descriptio i the Eistei frame is more ituitive, due to the caoical form of the scalar field s kietic term ad the miimal couplig betwee the field ad the gravity. The oly exceptio is the ϕ limit, which usually leads to V due to the ϕ 2 term i the potetial. This ifiity comes from the sigularity of the Eistei frame metric tesor ad it does ot appear i the Jorda frame aalysis. For the vacuum model with f(r) = R + αr, (α > 0, > 0), (2.14) 3

5 oe obtais 3H 2 (1 + αr 1 ) = 1 2 ( 1)αR 3( 1)αHR 2 Ṙ. (2.15) I the regime F 1, ɛf > 1 oe fids [8] Ḣ H 2 ɛ, ɛ = 2 ( 1)(2 1), H 1 ɛt, a t1/ɛ. (2.16) The same results would be obtaied for the power-low iflatio i GR frame (for miimally coupled scalar field) with the potetial V = V 0 e λφ, where ɛ = λ 2 /2. However, such a model would geerate differet spectrum of primordial ihomogeeities. Let us ote that the ɛ ca be iterpreted as a slow-roll parameter oly for F 1 ad ɛ > 1/F. The secod coditio comes from the fact, that the correctio to the equatio of motio comig from the GR low-eergy limit shall be of the order of 1/F R/f(R). The slow-roll parameter will be deoted as ɛ H ed defied by ɛ H := Ḣ H 2 ɛ + 1 F. (2.17) The iflatio eds whe ɛ H 1, which happes i the ɛf < 1 regime. Thus at the momet of the ed of iflatio oe fids ϕ 1 for all realistic values of. Besides ɛ H let us defie slow-roll parameters ɛ F ad η F by ɛ F := F 2HF, η F := F HF. (2.18) To obtai iflatio oe must satisfy ɛ H, ɛ F, η F 1. To satisfy this coditio i the ɛf > 1 limit we eed ɛ < 1 > 1 2 (1 + 3). To satisfy Ḣ < 0 ad H > 0 oe eeds < 2. Thus the allowed rage for is give by [8] [ 1 2 (1 + ] 3), 2. (2.19) Let us check whether the assumptio F 1 is valid durig the last e-foldig of iflatio, durig which the primordial ihomogeeities were geerated. The evolutio of a slow-roll parameters (ad their ɛf > 1 limit) as a fuctio of time for differet values of has bee preseted at fig. 1, 2, 3. Values of the α parameter have bee chose to satisfy ormalisatio of primordial curvature perturbatios at e-folds before the ed of iflatio. To obtai the umber of e-folds let us ote that for the slow-roll approximatio of Eq. (2.5,2.6,2.14) oe fids ( ) N Ñ 3 3 ϕ(2 ) + 2( 1) log (ϕ) + 4 4(2 ) log, (2.20) where N ad Ñ are the umber of e-folds i Jorda ad Eistei frames respectively. Usually the log(ϕ) term is subdomiat ad oe ca write ϕ(n) 1 ( 2 2 (2 e 4(2 ) N) ) 3. (2.21) 4

6 Ε, Ε H, 1 F Α F Ε 1 F H H 2 Ε Ε H, Ε F, Η F Α Η F Ε H Ε F 1 F t t Figure 1. Aalytical results versus umerical simulatio of slow-roll parameters ad their ɛf > 1 limit. Dots correspod to N = 60 ad N = 50 (left ad right dots respectively). At the momet of the horizo crossig oe obtais ɛf 1, so oe ca use the aalytical solutio from the Eq. (2.16) to describe the evolutio of space-time durig that period. Ε, Ε H, 1 F Α F 1 F Ε H H 2 Ε Ε H, Ε F, Η F Α Η F Ε H Ε F 1 F t t Figure 2. Aalytical results versus umerical simulatio of the slow-roll parameters ad their ɛf > 1 limit. Dots correspod to N = 60 ad N = 50 (left ad right dots respectively). At the momet of the horizo crossig the coditio ɛf > 1 is satisfied ad oe ca use the aalytical solutio from the Eq. (2.16) to describe the evolutio of space-time durig that period. This result has bee obtaied usig the slow-roll approximatio ad its accuracy is of order of few e-foldigs. The Eistei frame scalar potetial for f(r) = R + αr as a fuctio of ϕ has a followig form ( V (ϕ) = α( 1)(α) ) 2 (ϕ 1) 2 1, (2.22) ϕ where the last term parametrizes the deviatio from the Starobisky potetial. Let us discuss case by case specific choices for the power i the last term i the cotext of the features of the potetial. a) To obtai the real values of the potetial for all ϕ together with the stable miimum at 5

7 Ε, Ε H, 1 F Α F 1 F Ε H H t Ε Ε H, Ε F, Η F Α Ε H Ε F 1 F Η F t Figure 3. Aalytical results versus umerical simulatio of the slow-roll parameters. Dots correspod to N = 60 ad N = 50 (left ad right dots respectively). At the momet of the horizo crossig oe obtais ɛf 1, so the aalytical solutio from the Eq. (2.16) caot be used to describe the evolutio of space-time durig that period. ϕ = 1 oe eeds 2 1 = 2l 2k + 1, (2.23) where k, l N ad l k. To satisfy these coditios oe eed to assume that = 2(1 2 a 10 b ), (2.24) where a, b N, a b ad a < b log 2 (10)+log 2 (3 3) 2. The last coditio is eeded i order to obtai accelerated expasio of the space-time. This case is preseted at the left pael of the Fig. 6. For this class of potetials the reheatig of the Uiverse takes place durig the oscillatios period, after the scalar field reach its miimum. b) The real values of the potetial for all ϕ ad a potetial without ay miimum is obtaied for 2 1 = 2l + 1 2k + 1, (2.25) where k, l N ad l < k. Potetial obtais egative values for ϕ < 1 ad it heads to for ϕ 0. This case is preseted at the middle pael of the Fig. 6. Such a potetial is o-physical ad without additioal terms it caot be used to geerate iflatio. c) I all other cases the potetial V becomes imagiary for ay ϕ < 1 ad it has o miimum. This case is preseted at the right pael of the Fig. 6. The iflatio eds with the reheatig of the Uiverse, which takes place whe the scalar field rolls towards ϕ = 1. Sice there is o oscillatio phase such a potetial requires strog coupligs betwee the auxiliary field ad matter fields. 2.1 The geeratio of primordial ihomogeeities The power spectrum of the Jorda frame primordial curvature perturbatios at the superhorizo scales has the followig form [8] ( ) P R 2H2 F H 2 3F. (2.26) 2 2π 6

8 Let us ote that the Eq. (2.2) ca be expressed as ɛ H = ɛ F (1 η F ), so for the slowroll approximatio oe fids ɛ H ɛ F. This approximatio is valid for 2 1, sice ɛ F = ( 1)ɛ H for ɛf > 1. Thus, the spectral idex s ad tesor to scalar ratio r are of the form of [8] s 1 4ɛ H + 2ɛ F 2η F 1 6ɛ H 2η F, r 48ɛ 2 F 48ɛ 2 H. (2.27) The ormalisatio of primordial ihomogeeities requires that P 1/2 R at the momet of 50 to 60 e-folds before the ed of iflatio. Oe ca use the ormalisatio of the power spectrum to obtai the realistic values of α. From Eq. (2.5,2.6,2.14,2.26) i the slow-roll regime oe fids the α = α(), which (for realistic values of ) is plotted at the Fig. 4 The ( s, r) plae (which describes the shape of primordial curvature perturbatios) for the R + αr model have bee preseted at the Fig. 5. Α 50, Α N N U eff φ Α 10 7 Β φ Figure 4. Left pael: The aalytical solutio for α as a fuctio of at the momet of N = 50 or N = 60 (solid gree ad dashed red lies respectively). Right pael: The U eff i the vacuum case (solid blue lie) ad for the dust with ρ M = (dotted red lie), ρ M = (thick dotted brow lie), ρ M = (dashed yellow lie). The result is preseted up to a additive costat. The miimum is gettig closer to ϕ = 1 as the ρ M starts to domiate over U(ϕ). r 50, r s 50, 1 s r =2 =1.99 =1.98 =1.95 =1.9 =1.85 = s Figure 5. Left ad middle paels: The aalytical solutio for r ad s as a fuctio of at the momet of N = 50 or N = 60 (solid gree ad dashed red lies respectively). Right pael: The (r, s ) plae for differet. Dark ad light blue lies represet 1σ ad 2σ regios of the BICEP2 results [6]. The R + αr model is iside the 2σ regio for (1.805, 1.845). For 2 the results are cosistet with the PLANCK data. For completeess of the descriptio of the R + αr model we preset i the Appedix A the aalysis of the fial stage of the evolutio of the field ear the miimum, i cases whe the miimum does exist. 7

9 Α 1 1 V φ φ Α 1 1 V φ φ Α 1 1 V φ φ Figure 6. All paels show the Eistei frame scalar potetial V (ϕ) i the R + αr model. The left pael describes the case with the stable miimum. Middle ad right paels show the Eistei frame scalar potetial V (ϕ) i the case of lack of the miimum. At the right pael the potetial obtais complex values for ϕ < 0. 3 The R + αr βr 2 model Deficiecies of the model discussed i the previous Sectio ca be bypassed by cosiderig further modificatio of the gravitatioal Lagragia. To geerate a miimum for the scalar potetial ad (as we will show) to geerate dark eergy as a relict of iflatio let us cosider more complicated form of f(r), amely f(r) = R + αr βr 2, (3.1) where α ad β are positive costats ad satisfies the Eq. (2.19). Let us require α 1, β 1 ad αβ 1. This meas that the αr R βr 2 durig iflatio, so the results obtaied i the sec. 2 are still valid. From ϕ = F = 1 + αr 1 β(2 )R 1 (3.2) oe fids R(ϕ) = ( 4(2 )αβ + (ϕ 1) 2 + ϕ 1 2α ) 1 1 Let us ote that R > 0 for ay value of ϕ. For ϕ 1 αβ oe obtais (3.3) ( ϕ 1 R(ϕ) α ) 1 1 αr βr 2 1 αβ ( ) ϕ (3.4) Thus, the βr 2 term does ot have ay ifluece o iflatio ad geeratio of the large scale structure of the uiverse. The Jorda ad Eistei frames scalar potetials look as follows U(ϕ) = 1 2 ( 1) ( αr (ϕ) + βr 2 (ϕ) ), V = 1 ϕ 2 U (3.5) The Eistei frame scalar potetial has a miimum at R mi = ( 1 + 4(2 )αβ 1 2(2 )α ) 1 1 ( (β) ) (2 )αβ, (3.6) 1 8

10 where the last term is the Taylor expasio with respect to beta. The miimum is slightly shifted with respect to R = 0, which is the GR vacuum case. Hece, this model predicts some amout of vacuum eergy. Aroud the miimum oe fids αr 1 mi αβ 1, βr1 mi 1 O(1), (3.7) which meas that the iflatioary term is egligible ad R βr 2. The existece of the αr term is still importat, sice it provides the miimum ad real values of a scalar potetial for all ϕ. Let us clarify that R mi is ot the miimal value of R. The Ricci scalar has o miimum, its miimal value is equal to 0 (at the ϕ limit) ad it cotiuously grows with ϕ. The value of V at the miimum for small values of β reads V (ϕ mi ) 8( 1) 2 (β) 1 ( αβ ) 1 2 β 1 1, (3.8) where ϕ mi := F (R mi ) 2 ( 1)(1 + 2αβ) is the value of ϕ at the miimum of V. The ad that we eed Eq. 3.8 meas that the eergy desity of the DE shall be of order of β 1 1 β 1 to fit the dark eergy data. A example of a potetial ad its miimum are plotted at the Fig. 7. The existece of a stable miimum is oe of the mai differeces betwee this model ad a R βr 2 dark eergy model, i which the auxiliary field rolls dow towards egative ϕ for small eergies [17]. The miimum of the Eistei frame potetial (visible at both paels of the Fig. 7) prevets the ϕ from obtaiig egative values for ay solutio with iflatioary iitial coditios. 3.1 Coditios for the f(r) dark eergy Let us check whether this model satisfies coditios poited out i the Ref. [8]. The R 0 deoted the value of the Ricci scalar today 1) To avoid the ghost state oe eeds which meas that R 0 > R(ϕ = 0) = F > 0 for R R 0, (3.9) ( 1 + 4αβ(2 ) 1 2α ) 1 1. (3.10) This coditio is satisfied for ay ϕ with iitial coditio ϕ > 0, due to the existece of the miimum of the Eistei frame scalar potetial. This is a advatage of this model comparig to the f(r) = R βr 2 oe, i which F < 0 i late times. 2) To avoid the egative mass square for a scalar field degree of freedom oe eeds F > 0 for R R 0, (3.11) which meas that R 0 > 0. This coditio is obviously satisfied, sice the uiverse at the preset times is filled with DE ad dust. 9

11 3) To satisfy cosistecy with local gravity costraits oe eeds f(r) R 2Λ for R R 0, (3.12) After the ϕ field is stabilised i its miimum it produces the vacuum eergy, which is a source of Λ. The o-zero values of R comes from radiatio ad dust produced durig the reheatig of the uiverse, as well as from the o-zero miimal value of the Ricci scalar. 4) For the stability ad the presece of a late-time de Sitter solutio oe eeds The r = 2 happes for 0 < RF F < 1 at r := RF f = 2 (3.13) R = R r = ( (2 ) αβ 1 2(2 )α At R = R r the RF /F takes the form of ) 1 1 (β) 1 1. (3.14) RF F (r = 2) = 1 ( ( ) αβ ( 1) ) (2 ) αβ, 2 + 8αβ (3.15) which lies i the rage of (0, 1) for cosidered regimes of α ad β. To see that clearly let us perform a Tylor expasio of the RF /F with respect to αβ. The oe obtais RF F (r = 2) 1 2 (2 ) + ( 1)2 αβ (3.16) Thus, this model satisfies coditios for a viable DE model V φ Α 10 7 Β V φ Α 10 7 Β φ φ Figure 7. Both paels show the Eistei frame scalar potetial V (ϕ) ad its miimum i the R + αr βr 2 model. Note that V (ϕ mi ) , which gives the o-zero vacuum eergy. 10

12 4 Numerical aalysis of the dark eergy model The o-zero value of the Eistei frame potetial at the miimum rises a possibility of obtaiig a realistic solutio to the dark eergy problem. To aalyse low eergy solutios of the Jorda frame equatios of motio let us use the umber of e-folds (defied by N := log(a)) as a time variable. The Eq. (2.5,2.6) read H 2 (ϕ NN + 3ϕ N ) + H N Hϕ N (ϕu ϕ 2U) = 1 3 (ρ M 3p M ), (4.1) H 2 = ρ M + U 3 (ϕ + ϕ N ), (4.2) where the idex N deotes the derivative with respect to N. Sice i the Jorda frame the Eq. (2.7) is satisfied oe fids ρ M = ρ I e 3(1+w)N, where w = p M /ρ M is a barotropic parameter. After the iflatio ϕ oscillates aroud ϕ mi ad reheats the uiverse by the particle productio. Thus, after oscillatios oe obtais the radiatio domiatio era, for which w = 1/3 ad ρ M 3p M = 0. The radiatio icreases the cosmic frictio term but does ot cotributes to the U eff, so the field is ot shifted from the miimum. However, durig the dust domiatio era the U eff is modified ad ϕ oscillates aroud ϕ = 1. The evolutio of ϕ ad ϕ N durig the dust/de domiatio era is preseted at the Fig. 8. The evolutio of the Hubble parameter ad ρ M /3 is plotted at the Fig. 9. We have assumed that the field starts from the ϕ 1 (which is the GR limit of the theory), but umerical aalysis shows that the late-time results do ot deped o iitial coditios. For istace the iitial value of the field i the Eistei frame miimum the oly differece is slightly loger period of oscillatios aroud ϕ = 1. As log as the ρ M domiates over the eergy desity of the Bras-Dicke field the field oscillates ad stabilises above the ϕ mi. Durig that period ϕ 1 1 ad the R term i f(r) domiates. Thus oe recovers the GR limit of the theory. Whe the dust becomes subdomiat the ϕ rolls to its miimum ad oe obtais the Dark Eergy with the barotropic parameter ω = 1. As show i the Fig. 9 the Hubble parameter obtais the costat value whe the auxiliary field is i its miimum. The H = cost implies that R = 12H 2 = 4ρ Λ, where ρ Λ is the DE eergy desity. While ϕ is i its miimum oe fids R = R mi, so at the late times oe fids ρ Λ = R mi /4 1 4 (β) 1 1 β 1 (4ρ Λ) 1. (4.3) The ρ Λ becomes costat eve before the DE domiatio era. Thus, the Eq. 4.3 shall be satisfied at the preset time. 5 Coclusios I this ote we have demostrated how to exted the f(r) = R + αr model to combie successfully both features of the observable Uiverse i a sigle framework: we show how to obtai successful iflatio ad the o-zero residual value of the Ricci scalar i a extesio of the Starobisky model. Our results ca be easily cosistet with PLANCK or BICEP2 data for appropriate choices of the value of. I this case the Eistei frame potetial of the auxiliary field has a miimum oly whe (2 )/( 1) is of the form of a fractio with eve ad odd umbers i the umerator ad the deomiator respectively. The potetial 11

13 jn j N N Figure 8. Numerical results of the evolutio of the Bras-Dicke field ϕ ad its derivative with respect to N as a fuctio of N. We have assumed = 1.95, α = 2 108, β = 10 30, ϕ(0) = 1 ad ϕn (0) = 0. The ρm is the dust with ρi The choice of iitial coditios ad parameters of the model is rather urealistic (to big β ad ρi ), but it illustrates the way of the field to its miimum. UHjL j + 3Hj jl2 4, ΡM j -13 log10 HHL H ΡM -18 3j N N Figure p 9. Left pael: umerical result of the evolutio of the Hubble parameter (gree dashed lie) ad ρm /3ϕ (red solid lie) as a fuctio of time. Note that the vacuum eergy of the Bras-Dicke field starts to domiate whe the field reaches it s miimum. Right pael: umerical result for the evolutio of dust ad auxiliary field s eergy desity (red ad gree lies respectively). The eergy desity of the ϕ obtais the barotropic parameter ωϕ ' 1 about 10 e-folds before the DE domiatio period. has a miimum at ϕ = 1 ad V (ϕ = 1) = 0, which meas that there is o vacuum eergy. For all other forms of (2 )/( 1) the miimum does ot exist ad the potetial goes to or becomes complex for ϕ < 1. I the sectio 3 we have geeralised this model ito R + αr βr2, with α 1 ad β 1. I this case the Ricci scalar is always bigger tha zero. The Eistei frame scalar potetial is real for all ϕ ad it has a miimum for all. The potetial has o-zero value at the miimum, which may become a source of DE. The value of the parameter α is set by the ormalisatio of primordial ihomogeeities, while the value of the parameter β ca be read from the measured value of the preset DE eergy desity. I the sectio 4 we have performed umerical aalysis of the late-time evolutio of the R + αr βr2 model with dust employed as a matter field. Durig the radiatio domiatio era the ρm 3pM = 0, so the effective potetial i the Jorda frame obtais its vacuum form. Thus the field holds ϕ = ϕmi. Durig the dust domiatio era oe fids 12

14 ϕ = F 1, which correspods to the GR limit of the theory. Whe matter starts to be subdomiat the ϕ rolls to its miimum i ϕ = ϕ mi 1. Eve before that momet the eergy desity of the auxiliary field becomes costat ad the eergy desity of the ϕ evolves like DE with barotropic parameter ω ϕ = 1. From the preset value of the DE eergy desity we have foud the value of the parameter β to be (4ρ Λ ) 1 /. 6 Ackowledgemets We would like to thak prof. Ma for useful discussios. This work has bee supported by Natioal Sciece Cetre uder research grat DEC-2012/04/A/ST2/00099 ad partially by research grat DEC-2011/01/M/ST2/02466, by NSFC (No ) ad the Fudametal Research Fuds for the Cetral Uiversity of Chia uder Grat No.2013ZM107. M.A. would also like to ackowledge Chia Postdoctoral Sciece Foudatio for fiacial support. A Oscillatios aroud the miimum Let us assume that the coditio (2.23) is satisfied. The the iflatio eds with the period of oscillatios of the scalar field aroud the miimum of its potetial. To obtai the approximate form of the potetial V from the Eq. (2.22) aroud the miimum ϕ = 1 oe shall express V i term of the Eistei frame field φ = 3/2 log(ϕ) ad perform a Tylor expasio aroud φ = 0. The the leadig term is equal to Durig the oscillatio phase oe obtais V (φ) 6 2( 1) 4( 1)α(α) 1 φ 1. (A.1) < φ 2 t > < V φφ >, (A.2) where < > deotes the average value over oe period of oscillatios. Thus, the effective barotropic parameter i the Eistei frame (deotes as ω φ ) is equal to ω φ = < p > 1 < ρ > = 2 < V φφ > < V > 1 2 < V φφ > + < V >, (A.3) where V φ = dv dφ. For the potetial of the form of (A.1) oe fids ω φ = Thus, durig the oscillatio phase the Eistei frame scale factor is proportioal to ã = t 2 3(1+ω φ) = t From the first Firedma equatio i the Eistei frame oe obtais (A.4) (A.5) < 3 H 2 >=< 1 2 φ2 t > + < V >= 1 2 < V φφ > + < V >, (A.6) where H := ã t /ã is the Eistei frame Hubble parameter. From Eq. (A.2,A.6) oe fids ( ) 1 ( ) 1 < φ > (α) [1 ( ) 1 ] 6 3 2, < ϕ >= exp t 2 3 α 2 3 α t 2 3 α 2. (A.7) By defiitio the Jorda frame scale factor is equal to a = ãϕ 1/2, which gives us the evolutio of a as a fuctio of the Eistei frame time variable. 13

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